Gradient flow of O(N) nonlinear sigma model at large N

We study the gradient flow equation for the O(N) nonlinear sigma model in two dimensions at large N. We parameterize solution of the field at flow time t in powers of bare fields by introducing the coefficient function Xn for the n-th power term (n = 1, 3, ··· ). Reducing the flow equation by keeping only the contributions at leading order in large N, we obtain a set of equations for Xn’s, which can be solved iteratively starting from n = 1. For n = 1 case, we find an explicit form of the exact solution. Using this solution, we show that the two point function at finite flow time t is finite. As an application, we obtain the non-perturbative running coupling defined from the energy density. We also discuss the solution for n = 3 case.